dx 2ax - x^2 = a^n ⁻¹ ( x a - 1 ) . Determine the value of n . You may use dimensional analysis to solve the problem.

Step-by-step solution

Let the dimension of x be [L] . The dimension of dx is [L] . For the term 2ax - x^2 to be dimensionally consistent, the dimension of a must be [L] . The dimension of the denominator 2ax - x^2 is [L^2] = [L] . Therefore, the dimension of the left-hand side is [L] [L] = [L^0] . On the right-hand side, the argument of the inverse sine function, x a - 1 , is dimensionless. The function ⁻¹ yields an angle, which is also dimensionless. Thus, the dimension of the right-hand side is [a^n] = [L]^n . Equating the dimensions